Circle of Fifths

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The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Circle of Fifths

 

a graphic diagram of keys showing how they are related. The keys of C major and A minor are placed at the top of the circle; moving clockwise from them the major and minor sharp keys are arranged in perfect fifths, and the flat keys are arranged moving counterclockwise. To construct the sharp and flat keys of all 12 tones of the octave, the movement in both directions must be full circle. Here each key will have an enharmonic equivalent. Since keys having more than seven sharps or flats are not used in reality, the sharp and flat keys close up into a common circle of fifths, joining with their enharmonic equivalents in its three lower links.

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
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It had simple appeal, but was based on a very elaborate thought process and, I remember her telling me, a "whacky diagram" relating to the musical 'circle of fourths' and Eastern philosophy.
The group [C.sub.12] has four generators: 1 and its inverse 11; and 5 and its inverse 7.(6) The sequence generated by i is simply the ascending chromatic scale, that by 11 the descending chromatic scale, while 5 and 7 generate the circle of fourths and circle of fifths, respectively (or, alternatively, the descending and ascending circle of fifths).
The whole array of key signatures can be constructed easily by adding a perfect seventh (11S) for sharps and a perfect sixth (9S) for flats in complete analogy with the use of the circle of fifths for sharps and circle of fourths for flats in the twelve-tone system.